Math Problem Statement
Solution
The given augmented matrix can be written as a system of equations by interpreting the rows as coefficients for variables , , and , with the final column representing the constants on the right-hand side of each equation.
Augmented Matrix:
1 & -3 & 0 & 400 \\ 0 & 1 & 0 & 150 \\ 2 & 4 & 1 & 200 \end{bmatrix}$$ ### Step-by-Step Conversion: 1. The first row represents: $$1x - 3y + 0z = 400$$ Simplified: $$x - 3y = 400$$ 2. The second row represents: $$0x + 1y + 0z = 150$$ Simplified: $$y = 150$$ 3. The third row represents: $$2x + 4y + 1z = 200$$ Simplified: $$2x + 4y + z = 200$$ ### Final System of Equations: $$\begin{aligned} x - 3y + 0z &= 400, \\ 0x + y + 0z &= 150, \\ 2x + 4y + z &= 200. \end{aligned}$$ Let me know if you'd like to see how this system can be solved! --- ### Follow-Up Questions: 1. How can you solve this system of equations using substitution or elimination? 2. What does the augmented matrix represent geometrically in terms of planes or lines? 3. How can row-reduction be used to simplify solving such matrices? 4. Can this system be represented and solved in a matrix form using technology? 5. How does the presence of a zero coefficient simplify solving the system? --- ### Tip: When converting an augmented matrix to a system of equations, always match each row to its respective variables and check coefficients carefully.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Matrix Conversion
Linear Systems
Formulas
Matrix to Equation Conversion: Row[i] => a_i*x + b_i*y + c_i*z = d_i
Theorems
Linear Independence
Matrix Representation of Linear Systems
Suitable Grade Level
Grades 10-12
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